When designing a rubber product, methods that employ 3-D Finite Element Analysis (FEA) methods to predict the elastic response thereof, and that employ such analysis and simulation results have already been implemented for tens of years. Energy constitutive equations are employed in FEA computation and in the analysis reflect a stress-strain relationship of a rubber material, migrate linear elasticity equations into Mooney-Rivlin equations, and recently non-linear constitutive equations are being introduced for regions of large deformation in such energy constitutive equations.
Moreover, recently with regard to such a rubber material constitutive equation, under the principles of lattice deformation theory developed with molecular statistical thermodynamics based on the extension of rubber molecular chains, there is a proposal for a constitutive equation that is capable of expressing temperature dependence, an important factor in the design of a lot of industrial rubber materials, such as tires, using a parameter with a physical meaning (Japanese Patent No. 4299735).
A summary of such a constitutive equation follows.
First, tensional stress in rubber is expressed by a shear elastic modulus G and a tension direction extension ratio λ using the following Equation (1).
                    σ        =                  2          ⁢                      G            ⁡                          (                              λ                -                                  1                                      λ                    2                                                              )                                                          (        1        )            
Moreover, the shear elastic modulus G of rubber is computable, as shown in the following Equation (2), by differentiating the Helmholz free energy A of statistical thermodynamics with respect to I1.
                              G          2                =                              ∂            A                                ∂                          I              1                                                          (        2        )            
The average energy Āα of a system in a thermodynamic equilibrium state is expressed by the following Equation (3).
                                          A            _                    α                =                                            1              Z                        ⁢                          ∑                                                A                  α                                ⁢                                  ⅇ                                                            -                      β                                        ⁢                                                                                  ⁢                                          A                      α                                                                                                    =                      -                                          (                                                                            ∂                      log                                        ⁢                                                                                  ⁢                    Z                                                        ∂                    β                                                  )                                            {                Γ                }                                                                        (        3        )            
β here is equivalent to 1/(kΔT), wherein k represents the Helmholz constant and ΔT represents the difference from a polymer glass transition temperature Tg to the measured temperature T. Moreover, Aα represents the energy level. Z is a distribution function for standardizing the energy of the system, and is expressed by the following Equation (4).
                    Z        =                                            ∑              α                        ⁢                          ⅇ                                                -                  β                                ⁢                                                                  ⁢                                  A                  α                                                              =                                    ∑              α                        ⁢                          ⅇ                              -                                  β                  ⁡                                      (                                                                  U                        α                                            -                                              TS                        α                                                              )                                                                                                          (        4        )            
Here, as expressed in the following Equation (5), Uα can be assumed to be equivalent to Hamiltonian H from a statistical thermodynamic perspective.Uα=H (Temp, Constraints)  (5)
Wherein, the Hamiltonian H is expressed as a function of temperature conditions and constraint conditions defining microstates in statistical thermodynamics.
Two states are set, a low temperature state and a high temperature state, as a model expressing temperature dependence of rubber molecules, and state numbers of rubber molecule called r1 and r2 are distributed in strain energy fields called I1a and I1b. This can be represented by the following Equation (6) when this is applied to express all of the states as a function.
                                                                        Z                =                                                                            ∑                      α                                        ⁢                                          ⅇ                                                                        -                          β                                                ⁢                                                                                                  ⁢                                                  A                          α                                                                                                      =                                                                                    ∑                        α                                            ⁢                                              ⅇ                                                  -                                                      β                            ⁡                                                          (                                                                                                U                                  α                                                                -                                                                  TS                                  α                                                                                            )                                                                                                                                            =                                                                                  ⁢                                                                                  ⁢                                                                  ⅇ                                                  β                          ⁢                                                                                                          ⁢                          TS                                                                    ⁡                                              (                                                                              ⅇ                                                          -                                                                                                                                    -                                                                          I                                                                              1                                        ⁢                                        a                                                                                                                                              -                                                                      I                                                                          1                                      ⁢                                      b                                                                                                        -                                  κ                                                                                                  k                                  ⁢                                                                                                                                          ⁢                                  Δ                                  ⁢                                                                                                                                          ⁢                                  T                                                                                                                                              +                                                      ⅇ                                                          -                                                                                                                                    -                                                                          I                                                                              1                                        ⁢                                        a                                                                                                                                              +                                                                      I                                    b                                                                                                                                    k                                  ⁢                                                                                                                                          ⁢                                  Δ                                  ⁢                                                                                                                                          ⁢                                  T                                                                                                                                              +                                                      ⅇ                                                          -                                                                                                                                    I                                                                          1                                      ⁢                                      a                                                                                                        -                                                                      I                                    b                                                                                                                                    k                                  ⁢                                                                                                                                          ⁢                                  Δ                                  ⁢                                                                                                                                          ⁢                                  T                                                                                                                                              +                                                      ⅇ                                                          -                                                                                                                                    I                                                                          1                                      ⁢                                      a                                                                                                        +                                                                      I                                                                          1                                      ⁢                                      b                                                                                                        -                                  κ                                                                                                  k                                  ⁢                                                                                                                                          ⁢                                  Δ                                  ⁢                                                                                                                                          ⁢                                  T                                                                                                                                                                    )                                                                                                                                                                    =                                  2                  ⁢                                                            ⅇ                                              β                        ⁢                                                                                                  ⁢                        TS                                                              ⁡                                          [                                                                                                    ⅇ                                                          κ                                                              k                                ⁢                                                                                                                                  ⁢                                Δ                                ⁢                                                                                                                                  ⁢                                T                                                                                                              ⁢                                                      cosh                            ⁡                                                          (                                                                                                                                    I                                                                          1                                      ⁢                                      a                                                                                                        +                                                                      I                                                                          1                                      ⁢                                      b                                                                                                                                                                        kΔ                                  ⁢                                                                                                                                          ⁢                                  T                                                                                            )                                                                                                      +                                                  cosh                          ⁡                                                      (                                                                                                                            I                                                                      1                                    ⁢                                    a                                                                                                  -                                                                  I                                                                      1                                    ⁢                                    b                                                                                                                                                              k                                ⁢                                                                                                                                  ⁢                                Δ                                ⁢                                                                                                                                  ⁢                                T                                                                                      )                                                                                              ]                                                                                                                                       (        6        )            
Wherein κ represents molecular recoil energy. Two states of different rubber molecule energy states are represented by r1=−1, r2=1, and the statistical thermodynamic average thereof is represented by <r1·r2>. This product expresses the two extreme end energy states, and when this product is +1 this represents rubber molecules being in the same state, and when this product is −1 this represents rubber molecules being in different states. Assuming that the strain energy field contribution is equal for all of the molecules, then by making I1=I1a=I1b, Equation (6) above can be expressed by the following Equation (7).
                    Z        =                                            ∑              α                        ⁢                          ⅇ                                                -                  β                                ⁢                                                                  ⁢                                  A                  α                                                              =                      2            ⁢                                          ⅇ                                  β                  ⁢                                                                          ⁢                  TS                                            ⁡                              [                                                                            ⅇ                                              κ                                                  k                          ⁢                                                                                                          ⁢                          Δ                          ⁢                                                                                                          ⁢                          T                                                                                      ⁢                                          cosh                      ⁡                                              (                                                  2                          ⁢                                                      β                            ·                                                          I                              1                                                                                                      )                                                                              +                  1                                ]                                                                        (        7        )            
Substituting Equation (7) in above Equation (3) expresses the rubber elastic modulus G by the following Equation (9).
                                          A            _                    α                =                                            1              Z                        ⁢                          ∑                                                A                  α                                ⁢                                  ⅇ                                                            -                      β                                        ⁢                                                                                  ⁢                                          A                      α                                                                                                    =                                    -                                                (                                                                                    ∂                        log                                            ⁢                                                                                          ⁢                      Z                                                              ∂                      β                                                        )                                                  (                  Γ                  )                                                      =                                          -                                                                            ∂                                                                                                                                  ∂                      β                                                        ⁡                                      [                                                                                            ⅇ                                                      κ                                                          k                              ⁢                                                                                                                          ⁢                              Δ                              ⁢                                                                                                                          ⁢                              T                                                                                                      ⁢                                                  cosh                          ⁡                                                      (                                                          2                              ⁢                                                              β                                ·                                                                  I                                  1                                                                                                                      )                                                                                              +                      1                                        ]                                                              -                              T                ·                S                                                                        (        8        )                                          G          2                =                                            ∂                                                A                  _                                α                                                    ∂                              I                1                                              =                                                                      ∂                  U                                                  ∂                                      I                    1                                                              -                              T                ·                                                      ∂                    S                                                        ∂                                          I                      1                                                                                            =                                          -                                                                            ∂                                                                                                                                                          ∂                        β                                            ⁢                                              ∂                                                  I                          1                                                                                                      ⁡                                      [                                                                                            ⅇ                                                      x                                                          k                              ⁢                                                                                                                          ⁢                              Δ                              ⁢                                                                                                                          ⁢                              T                                                                                                      ⁢                                                  cosh                          ⁡                                                      (                                                          2                              ⁢                                                              β                                ·                                                                  I                                  1                                                                                                                      )                                                                                              +                      1                                        ]                                                              -                              T                ·                                                      ∂                    S                                                        ∂                                          I                      1                                                                                                                              (        9        )            
Considering a stretched molecule, the second term of the entropy term in Equation (9) is expressed using a Langevin function by the following Equation (10).
                              T          ·                                    ∂              S                                      ∂                              I                1                                                    =                              -            v                    ⁢                                          ⁢                      RT            ⁡                          [                                                1                  2                                +                                                      3                                          50                      ⁢                                                                                          ⁢                      n                                                        ⁢                                      (                                                                  3                        ⁢                                                  I                          1                                                                    -                                              2                        λ                                                              )                                                  +                                                      297                                          6125                      ⁢                                                                                          ⁢                                              n                        2                                                                              ⁢                                      (                                                                  5                        ⁢                                                  I                          1                          2                                                                    -                                              4                        ⁢                                                  I                          2                                                                    -                                              4                        ⁢                                                                              I                            1                                                    λ                                                                                      )                                                              ]                                                          (        10        )            
Wherein n is defined as the number of links of a statistical molecule chain between cross-linked points.
Accordingly, Japanese Patent No. 4299735 proposes a constitutive equation that is capable of representing temperature dependence and employs a parameter with physical meaning.
Moreover, as illustrated in FIG. 6, a strain energy function (solid line) disclosed in Japanese Patent No. 4299735 is able to replicate test values (plots) over a wide range of extension ratios and temperature regions.